Who is King Lemuel in Proverbs 31? I did a search for King Lemuel in my entire library of encyclopedias, commentaries, Bible dictionaries, etc. and there is very limited information on him. Some see him as a descendent of Ishmael depending on how one answers the previous question on Proverbs 31.
Not that I have a whole lot to add because as you have already noted there is not much to go on as to Lemuel might have been.
Longman in his Dictionary of the Old Testament points out a very old issue that dates back to the time of the Septuagint: 1) Some Jewish sources have Agur and Lemuel as noms de plume (a Latin phrases for alias) for Solomon. That is why the LXX and the Latin Vulgate do not even recognize these two terms as proper names at all. If this is the case then a very interesting thing emerges from the text -- this wisdom is coming into the text via Bathsheba because in this case Lemuel is another name for Solomon. Bathsheba would have had a unique contribution to the issues found in this chapter. This would have a powerful impact on the text but it depends to a large extent on Jewish Tradition which here is divided. This is very sparse material, Longman mentions the noms de plume aspect but not the significance of it as identifying the mother. After finding this I did a quick search and even the Wiki page for Lemuel mentions that the mother of Proverbs 31 may have been Bathsheba.
2) Jewish Tradition also has argued that he was king of Massa which as you pointed out was a descendant of Ishmael. The term Massa comes into the picture through the Hebrew as a change in the punctuation. That term is rendered in many English versions as "the prophecy" (KJV) or "an oracle" (NASB) instead of the title of the place Lemuel was king of as this rendering, "Lemuel King of Massa."
A useful resource aside from the tools I have in Logos that can be helpful from time to time is the Jewish Encyclopedia site that is online. There are a lot of references to Lemuel on that site. After a basic search one of the more interesting ones is that Massa was related to the Kedemites, who were known for their wisdom.
Here is what the one entry suggested about Lemuel (following the #2 option above):
The Kedemites enjoyed among the Israelites a great reputation for wisdom. Not only does David quote a Kedemite proverb which he characterizes as such, but the wisdom of the Kedemites is rated only lower than Solomon's though higher than that of the Egyptians (I Kings 5:10), and Isaiah represents the Egyptian king's wise men as seeking to impress him by claiming descent from sages of Kedem (this, not "of old," is the meaning of qedem in Isa. 19:11). A wise instruction by the mother of a Kedemite king, *Lemuel , to her royal son is preserved, according to the superscription, in Proverbs 31:1–9; for Massa is the name of an Ishmaelite tribe (Gen. 25:14; on the Aramaizing diction, see *Job ). *Agur son of Jakeh was doubtless of the same nationality as Lemuel, according to Proverbs 30:1, where ha-massa'i, "the Massaite," is to be read. The reputed wisdom of the Edomites (included among the Kedemites in Isa. 11:14) is alluded to in Jeremiah 49:7; Obadiah 7 end, 8.
Here is another useful link: https://claudemariottini.com/2009/05/18/who-was-king-lemuel/